Problem 4 - Introductory and Intermediate Algebra for College Students 4th

Question

Write a positive or negative integer that describes each situation. A loss of \(\$ 12,500.00\)

Step-by-Step Solution

Answer

The situation can be described by the negative integer -12500.

Step 1: Analyzing the situation

In this situation, there is a loss of \$12,500.00. We need to describe this situation using a positive or negative integer.

Step 2: Representing the loss

In mathematical terms, a loss or decrease is typically represented by a negative number, as it involves 'taking away' from an original amount, reducing it.

Step 3: Finalizing the integer

Hence, this loss of \$12,500.00 would best be represented by the negative integer, -12500.

Key Concepts

Positive and Negative Integers

Understanding positive and negative integers is foundational in mathematics, especially when representing real-world scenarios such as profit and loss or temperature variations. Integers include all whole numbers and their opposites. For instance, the number 3 is a positive integer, while -3 is a negative integer. The sign of an integer determines whether it is positive or negative.

Positive integers are used to represent a gain or increase in value, like depositing money into a bank account. In contrast, negative integers indicate a decrease or loss, like withdrawing money from that account. The number 0 is neutral; it's neither positive nor negative, but it's an integer. Visualizing these numbers on a number line helps in understanding their order and the concept of absolute value, which is the distance between the number and zero without considering direction.

• Positive integers: +1, +2, +3, ... (located to the right of 0 on the number line)
• Negative integers: -1, -2, -3, ... (located to the left of 0 on the number line)
• The absolute value of -5 is 5, just as the absolute value of +5 is 5

Integer Representation

When it comes to integer representation, the rules are straightforward. Each number is considered in relation to zero on a number line. In this context, the negative sign (-) is not just an ordinary dash; it represents the idea of 'opposite to' or 'less than zero'. Numbers without a sign are assumed to be positive. It's essential to understand that both symbols + and - do not merely represent operations; they are part of the integer's identity, showing its position relative to zero.

To visualize integer representation, consider temperature. A positive reading indicates degrees above freezing, while a negative reading signifies degrees below freezing. Similarly, in financing, assets and profits are mapped by positive numbers, whereas debts and losses are charted by negative numbers.

• '+7' indicates a point seven units to the right of zero
• '-3' indicates a point three units to the left of zero
• The transition from positive to negative, or vice versa, is marked by passing through zero

Mathematical Description of Loss

A mathematical description of loss is a practical interpretation of negative integers. It's used in finance, economics, and everyday life to quantify the subtraction of value or resources. The concept of loss is key to understanding a broad range of subjects from simple budgeting to complex economic theories.

When representing a loss numerically, a negative sign precedes the number. This subtraction from a base value, whether it's money, points in a game, or stock values, is a critical aspect of calculation and analysis. The use of negative numbers highlights the decrease from a previous state or a benchmark.

• In a mathematical description of a financial loss of \( \$12,500.00 \), we write it as '-\$12,500' or simply '-12500'
• A sports team moving 3 positions down in league rankings might be depicted as a change of '-3' in the standings
• For temperatures, a drop of 10 degrees might be expressed as '-10' degrees relative to a certain reference point